行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}@{}}{-}1 & {-}1 & 1 & 2 \\ 0 & 0 & {-}2 & {-}2 \\ 1 & 1 & 2 & 1 \\ 1 & 0 & 0 & {-}2\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 & 2 \\ 0 & 0 & {-}2 & {-}2 \\ 1 & 1 & 2 & 1 \\ 1 & 0 & 0 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [2, -2, 1, -2]

\( \qquad\quad = (-1)^{4-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 0 & {-}2 \\ 1 & 1 & 2 \\ 1 & 0 & 0\end{array}\,\right| + (-1)^{4-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 1 & 1 & 2 \\ 1 & 0 & 0\end{array}\,\right| + (-1)^{4-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 0 & 0 & {-}2 \\ 1 & 0 & 0\end{array}\,\right| + (-1)^{4-4}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 0 & 0 & {-}2 \\ 1 & 1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times2\times2 + (-1)^{4-2}\times(-2)\times(-3) + (-1)^{4-3}\times1\times2 + (-1)^{4-4}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 0 & {-}2 \\ 1 & 1 & 2 \\ 1 & 0 & 0\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, 2, 0]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 1 \\ 1 & 0\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times(-1) + (-1)^{3-2}\times2\times0 \)

\( \qquad\quad = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 1 \\ 1 & 0\end{array}\,\right| = 1\times0 - 1\times1 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 1 & 0\end{array}\,\right| = 0\times0 - 0\times1 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 1 & 1 & 2 \\ 1 & 0 & 0\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 0]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 1 \\ 1 & 0\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-1) + (-1)^{3-2}\times2\times1 \)

\( \qquad\quad = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 1 \\ 1 & 0\end{array}\,\right| = 1\times0 - 1\times1 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 1 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times1 = 1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 0 & 0 & {-}2 \\ 1 & 0 & 0\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, -2, 0]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 1 & 0\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times0 + (-1)^{3-2}\times(-2)\times1 \)

\( \qquad\quad = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 1 & 0\end{array}\,\right| = 0\times0 - 0\times1 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 1 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times1 = 1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & 1 \\ 0 & 0 & {-}2 \\ 1 & 1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, -2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 1 & 1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 1 & 1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times0 + (-1)^{3-2}\times(-2)\times0 + (-1)^{3-3}\times2\times0 \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 1 & 1\end{array}\,\right| = 0\times1 - 0\times1 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 1 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times1 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 0 & 0\end{array}\,\right| = (-1)\times0 - (-1)\times0 = 0 \)